HomeConcept libraryMechanicsMotion and Circular Motion

MechanicsIntroStarter track

Concept module

Projectile Motion

Launch a projectile, watch the trajectory form, and connect the range, height, and component motion to the launch settings.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 2.60 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s2.60 s

Projectile motion

Drag the launch vector to set angle and speed.

Graphs

Switch graph views without breaking the live stage and time link.

Trajectory

Shows the full path of the projectile through space.

horizontal distance (m): 0 to 40height (m): 0 to 12

Trajectory

Hover or scrub to link the graph back to the stage.horizontal distance (m) / height (m)

Controls

Adjust the physical parameters and watch the motion respond.

Launch speed

v_{0}

18 m/s

Controls the size of the initial push.

Launch angle

θ

45°

Controls how much of the launch speed points upward.

Gravity

g

9.8 m/s²

Controls how strongly the projectile falls back down.

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Graph readingPrompt 1 of 2

Notice that the trajectory graph and the stage are the same path shown in two representations.

Try this

Hover or scrub the trajectory graph and watch the projectile move to the matching point in the stage.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

v_{0}

Launch speed

18 m/s

Sets the size of the initial push, so the arc stretches farther and the velocity components start larger.

Graph: TrajectoryGraph: Component motionGraph: Velocity componentsOverlay: Velocity vectorOverlay: Component vectors

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompt as a short guide while you change the launch. The strongest prompt should point you toward a pattern that the current stage or graph can actually show.

Graph readingPrompt 1 of 2

Graph: Trajectory

Notice that the trajectory graph and the stage are the same path shown in two representations.

Try this

Hover or scrub the trajectory graph and watch the projectile move to the matching point in the stage.

Why it matters

It turns the graph from an appendix into a spatial description of the same flight.

Graph: TrajectoryOverlay: Apex markerOverlay: Range marker

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Velocity vector

Shows the projectile's instantaneous velocity.

What to notice

The vector tilts downward as gravity bends the path.

Why it matters

It links the launch settings to the moving projectile.

Control: Launch speedControl: Launch angleGraph: TrajectoryGraph: Velocity componentsEquation

v_{0}

Equation

θ

Challenge mode

Turn the same launch controls and time rail into compact aiming tasks. The checklist reads the real trajectory, not a separate answer key.

0/2 solved

ConditionStretch

0 of 4 checks

Freeze the apex

From Earth shot, pause exactly at the top of the arc where the vertical velocity has dropped to zero but the projectile is still high above the ground.

Inspect timeGraph-linkedGuided start2 hints

Suggested start

Pause the motion and scrub until the vertical-velocity curve crosses zero.

Pending

Open the Velocity components graph.

Trajectory

Pending

Pause into inspect mode.

live

Pending

Bring

v_{y}

close to

0 m/s

12.73 m/s

Pending

Keep the projectile height between

8.0

and

8.6 m

0 m

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the projectile has traveled 0 m horizontally and 0 m vertically. The launch angle is 45°, the predicted range is 33.06 m, and the maximum height is 8.27 m.

Equation detailsDeeper interpretation, notes, and worked variable context.

Horizontal position

Horizontal motion with constant velocity when drag is ignored.

x (t) = v_{0} cos (θ) t

v_{0}

Launch speed 18 m/s

θ

Launch angle 45°

Vertical position

Vertical motion pulled downward by gravity.

y (t) = v_{0} sin (θ) t - \frac{1}{2} g t^{2}

v_{0}

Launch speed 18 m/s

θ

Launch angle 45°

g

Gravity 9.8 m/s²

Flight time

The time before the projectile returns to the launch height.

T = \frac{2 v _{0} sin ( θ )}{g}

v_{0}

Launch speed 18 m/s

θ

Launch angle 45°

g

Gravity 9.8 m/s²

Range

The horizontal distance traveled when launch and landing heights are the same.

R = \frac{v _{0}^{2} sin ( 2 θ )}{g}

v_{0}

Launch speed 18 m/s

θ

Launch angle 45°

g

Gravity 9.8 m/s²

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 2 of 30 / 3 complete

Motion and Circular Motion

Earlier steps still set up Projectile Motion.

1. Vectors and Components2. Projectile Motion3. Uniform Circular Motion

Previous step: Vectors and Components.

Start from track beginning

Short explanation

What the system is doing

Projectile motion is what happens when an object gets an initial push and then gravity takes over. The horizontal and vertical parts of the motion separate cleanly, which makes the path predictable.

That separation is what makes the module useful. You can change launch speed, angle, and gravity, then watch the trajectory and its component graphs update together.

Key ideas

01The horizontal speed stays constant when drag is turned off.

02A 45 degree launch gives a long range only when the launch and landing heights match.

03Gravity changes the flight time and the arc, even if the launch speed stays fixed.

Live worked example

Solve the exact state on screen.

Solve the launch you are actually viewing. The givens and substitutions follow the real controls, and time-based examples follow the current inspected time unless you freeze them.

Live valuesFollowing current parameters

For the current launch settings, what horizontal range does the equal-height formula predict?

v_{0}

Launch speed

18 m/s

θ

Launch angle

45 °

g

Gravity

9.8 m/s²

1. Identify the relation

For equal launch and landing height, use

R = \frac{v _{0}^{2} sin ( 2 θ )}{g}

2. Substitute the current launch values

R = \frac{( 18 ) ^{2} sin ( 2 \cdot 4 5 ^{\circ} )}{9} .8

3. Compute the range

With

sin (2 θ) = 1

, the predicted range becomes

R = 33.06 m

Predicted range

R = 33.06 m

This angle keeps a fairly balanced split between horizontal reach and airtime, so the predicted range stays strong.

Common misconception

A steeper launch angle always gives a longer range.

Range depends on both horizontal and vertical components, not angle alone.

At the same launch speed, very steep angles waste horizontal speed and very shallow angles shorten flight time.

Mini challenge

If you keep speed fixed and lower the launch angle, what happens to the trajectory and the flight time?

Prediction prompt

Predict whether the arc gets taller or flatter.

Check your reasoning

The arc gets flatter and the flight time usually decreases.

A smaller angle sends more of the launch speed sideways and less upward, so gravity brings the object back down sooner.

Quick test

Compare cases

Question 1 of 4

Use the trajectory, the component graphs, and the launch variables together. Each question is meant to check whether you can explain the motion, not just name a formula.

At the same launch speed with equal launch and landing height, how do $3 0^{\circ}$ and $6 0^{\circ}$ launches compare?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a projectile launched from a point on the ground and traced through the air by gravity alone. The vector overlays can show the current velocity and its horizontal and vertical pieces.

Changing speed, launch angle, or gravity immediately reshapes the flight path, the landing point, and the component graphs.

Graph summary

The trajectory graph shows the curved flight path from launch to landing.

The component graphs show that horizontal motion stays steady while vertical motion bends under gravity.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Launch impulse viewMechanics

Projectile Motion

See each variable before you move it.

Velocity vector

Freeze the apex

Motion and Circular Motion

What the system is doing

Solve the exact state on screen.

For the current launch settings, what horizontal range does the equal-height formula predict?

At the same launch speed with equal launch and landing height, how do $3 0^{\circ}$ and $6 0^{\circ}$ launches compare?

Keep moving through the concept map.

Momentum and Impulse

Vectors and Components

Gravitational Fields

Projectile Motion

See each variable before you move it.

Velocity vector

Freeze the apex

Motion and Circular Motion

What the system is doing

Solve the exact state on screen.

For the current launch settings, what horizontal range does the equal-height formula predict?

At the same launch speed with equal launch and landing height, how do 30∘ and 60∘ launches compare?

Keep moving through the concept map.

Momentum and Impulse

Vectors and Components

Gravitational Fields

At the same launch speed with equal launch and landing height, how do $3 0^{\circ}$ and $6 0^{\circ}$ launches compare?